A Grey Anatomy: The Axon

This is the third in a series of occasional postings digger deeper into the brain’s structure and functioning. This one looks at the axon – the central cable-like structure within a neuron that, put simply, passes electrical signals down from the dendrites (inputs of the neuron) , via the ‘axon hillock’ at one end of the axon, to the ‘axon terminal’ (outputs of the neuron) at the other end of the axon.

Previous posts have looked at:

  1. ion channels and how they are similar to transistors, and
  2. building transistor functions from biological components.

In this post, the comparison with electronics is stronger since electrical circuit theory is used to model conduction down the axon. It provides a simple introduction to the Hodgkin-Huxley model. In so doing:

  • It explains how an action potential arises.
  • It explains the effect of myelination – the sheathing of the axon  in myelin, which is what gives these axons a white appearance hence the term white matter as opposed to grey matter.
  • It explains saltatory conduction along myelinated axons, and why an action potential can travel down such axons much faster and much further than down unmyelinated axons.

Cable Theory

Cable Theory uses mathematical models originally developed for electrical cables such as trans-continental telegraphic cables to model electrical conduction within neurons.

Below is a piecewise model of a such a transmission line in electrical engineering. We can view the long cable as being built up from many smaller sections of the cable, each  having the circuit diagram as shown. The components are:

Piece-wise linear model of electrical transmission line

Piece-wise linear model of electrical transmission line

  • Rs: series resistance,
  • Ls: series inductance,
  • Rp: parallel resistance,
  • Cp: parallel capacitance.

This is a general model that can also be applied to ‘co-axial cable’ such as that which runs from a home satellite dish to a TV set-top box.  For such a cable, the voltage here is the voltage between the inner core conductor and  the ‘ground’ shielding.

Because we are interested in the behaviour of the cable rather than an arbitrarily selected small section of it, we can replace actual resistance with ‘resistance per unit length’ e.g. Ohms per metre and similarly for capacitance and inductance.

A neuronal fibre, i.e. an axon, can be modelled in the same way as an unshielded electrical wire – in which there is no Ls series inductance component:

Piecewise linear model of an neuronal fibre (axon)

Piecewise linear model of an neuronal fibre (axon)

  • Rl: longitudinal resistance per unit length,
  • Rm: membrane resistance per unit length,
  • Cm: membrane capacitance per unit length.

The voltage is that across the membrane, between ‘inside the neuron’ and ‘outside the neuron’. A voltage difference is sometimes called a potential difference and this voltage is what is being referred to in an action potential. An action potential is a signal with a voltage that is large enough to propagate down the axon and thus trigger an action. More on this later.

Elementary formulae for the circumference and area of a circle mean that:

  • Rl is proportional to πr2 (fatter axons have lower longitudinal resistance),
  • Rm is inversely proportional to 1/2πr (fatter axons have a larger surface area to leak charge),
  • Cp is proportional to 2πr (fatter axons have a larger surface area to hold charge),

where r is the radius of the axon.

Solving various equations eventually leads to a formula for the voltage at a distance x into the axon being V(x)=V0.ex where λ is the ‘length constant’, λ=√(r/2).√(Rm/Rl). This exponentially-decaying type of formula should look familiar to electrical engineers. There are further similarities:

  • the voltage has decayed to 37% of the original (V0) after 1 length constant.
  • cable theory also an RC ‘time constant’: τ=Rm.Cm.

Note that fatter axons have a larger length constant so signals decay less quickly.

The Myelination of Axons

Myelin is an insulating material which is used to insulate axons, resulting in signal propagation down axons that can travel much faster and much further.

Axons can be either myelinated (in which case they are myelinated along their entire length) or unmyelinated. Since myelin appears white, it basically insulates axons in the white matter parts of the brain but not in the grey matter.

In the central nervous system (the brain):

    • Almost all axons with diameters greater than 0.2 μm are myelinated. The ratio between axon diameter and that of the total nerve fibre (axon and myelin) is 0.6–0.7, i.e. the myelin increases the diameter by about 50%.

Neuron with its axon myelinated by oligodendrocytes (Credit: Spinal Research)

  • Oligodendrocytes cells produce myelin and  ‘reach out’ to insulate (approx. 1 μm) sections of many (maybe 50)  nearby axons. Oligodendrocytes (Greek: ‘few+tree+glue’, cf. oligarchy and dendrochronology) are a type of glial (glue) cell in the brain that provides structural support for the neurons but much more – in this case, producing the myelin.

In the peripheral nervous system (nerves around our body):

  • Each myelin sheath has a Schwann cell to produce its myelin.
  • The length of each myelin sheath is approximately 1 mm with 1 μm-long gaps between adjacent sheaths, called the nodes of Ranvier.

Myelin decreases the capacitance across the cell membrane by a factor of about 5,000 and increases the resistance 50-fold. Using these values increases the time constant (τ=Rm.Cm) for conduction down the axon by a factor of 100 and increases the length constant by a factor of 7. Since myelinated axons generally have  a larger radius than unmyelinated ones and, as has been shown above, axons with a larger radius have a larger time constant, the speed effect of myelination is even greater than the calculated 100.

But there is a more significant factor involved with myelinated axons that explains how conduction can travel so far as well as so fast. Exponentially-decaying signals are not going to enable communication across the corpus callosum between the hemispheres or throughout the peripheral nervous system. That special factor is salutatory conduction, but before explaining that, we must understand how signals propagate along unmyelinated axons in more detail.

The Hodgkin–Huxley model

A neuron is a biological cell, with a membrane separating the insides from the outsides. For the axon, this part of the membrane is a long tube called the axolemma. Within the membrane are many enzymes (large biological molecules), such as:

  • Ion pumps, such as the sodium-potassium pump
  • Voltage-gated ion-channels: the flow of a particular ion through the membrane is controlled by the voltage across the membrane.
  • Ligand-gated ion-channels: the flow of a particular ion through the membrane is controlled by the whether a neurotransmitter molecule has locked onto the ion-channel or not.
  • Leak channels: particular ions flow through the membrane regardless of any controlling factor (the membrane has some ion permeability).

We can form a circuit diagram of an axon by connecting the various enzymes together electrically. Crudely we have the components:

Electrical analogy of enzymes within the axon membrane

Electrical analogy of enzymes within the axon membrane

  • ‘I’: An ion pump is analogous to a current source.
  • ‘V’: A voltage-gated ion-channel is analogous to a voltage-controlled current source (such as a transistor).
  • ‘L’: A ligand-gated ion-channel is analogous to a photodiode controlling a transistor. The generated current is controlled by some signal carrier (neurotransmitter is analogous to light) that is different from the normal charge carriers in the circuit (ions analogous to electrons).
  • ‘R’: A leak channel is analogous to a resistor, plus
  • ‘C’: The capacitance caused by the difference in numbers of charge carriers on either side of the cell membrance is also included.

The electrical components shown above are normally assumed  to be idealised, linear components. But in the biological case, this is certainly not the case. From the ‘actual’ circuit above, we can create a simplified lumped linear model as shown below which is the basic Hodgkin-Huxley model:

Chain of Hodgkin-Huxley models

Chain of Hodgkin-Huxley models

  • lumped: all the components of the same type are lumped together e.g. 1000 resistors of resistance R is replaced by a single resistor or resistance R/1000.
  • linear: simple Ohm’s law (V=I.R) applies for resistors and this has been achieved by putting in voltage source (battery symbols) to linearize them. Controlled current sources have a simple controlling formula e.g. I=g.V where g is a constant.

Hopefully it can be seen how the Hodgkin-Huxley model maps onto the cable theory model shown earlier, in which various controlled-current sources are further lumped into the Rp ‘membrane resistance per unit length’ metric.

Note that current flow around neurons is more complex than in the conventional electrical domain:

  • In electrical conductors, current flows solely due to electrons.
  • In electrical semi-conductors, current flows due to electrons (in the conducting band) and  holes (in the valence band).
  • In biophysical cells, current flows due to Potassium (K+), Sodium (Na+), Calcium (Ca2+) and Chlorine (Cl) ions.

So if, for example, all the K+ ions in a locality are inside the cell, there is little opportunity for a voltage-gated K+ channel to bring more K+ ions into the cell, regardless of the membrane voltage.

Action Potential

We are now in a position to explain the membrane voltage spikes, called action potentials, that occur at the axon hillock and which can then propagate down the axon.

This arises because of the following sequence of actions.

Action potential sequence

Action potential sequence

1: Resting Potential / Equilibrium

The membrane is an equilibrium. Sodium-Potassium ion pumps have repeatedly brought in 2 K+ ions in exchange for 3 Na+ ions and Potassium leak channels have let many Na+ ions escape. Having much more charge outside of the cell than inside leads to a membrane potential of -70mV. This non-zero voltage means that the membrane has been polarized.

2: Excitation

For reasons which will be explained in a later posting, input signals from other neurons causes the membrane potential at the axon hillock to exceed -55mV.

3: Depolarization

At this membrane potential, Na+ channels open letting Na+ ions into the cell. This results in the membrane potential increasing further, allowing the Na+ channels to let still more ions through. This positive feedback results in a very sharp increase in membrane potential, towards and past 0mV.

4: Overshoot

All other things being equal, the membrane potential would continue to let Na+ ions in until it reached  +58mV. But before that happens, the K+ channels start to open. What K+ ions there still are within the cell now get pushed out, reducing the membrane potential. The membrane potential reaches a maximum around 40mV.

5: Repolarization

Positive feedback now takes hold for the K+ channels, causing the membrane potential to fall sharping towards and then past the resting potential value of -70mV. All other things being equal, the membrane potential would continue to push K+ ions out until it reached  -93mV.

6: Hyperpolarization / Undershoot

Although the membrane potential is the same as the resting potential at -70mV, ion concentrations are very different. The axon locality is depleted of K+ ions but has a large concentration of Na+ ions. In contrast, when the axon locality is in its resting state, the cell is very depleted of Na+ ions and somewhat depleted of K+ ions.

Action potential opening and closing of voltage-gated ion channels (credit: Washington University)

There is a then gradual return to the equilibrium state through the action of the Sodium-Potassium ion pumps and Potassium leak channels, but not before the membrane potential has bottomed out at around -85mV.

Absolute Refractory Period

During most of the refractory period, a new excitation at the axon hillock is incapable of leading to a new action potential because the ion concentrations outside that per of the cell are not high enough for positive feedback to occur. This period is called the ‘absolute refractory period’ as new action potentials are absolutely not possible and begins at repolarisation when Na+ channels are inactivated.

Relative Refractory Period

A bit before the equilibrium state is reached, it again becomes possible for positive feedback to occur, although it takes an excitation charge greater than that normally to do so. This period is called the ‘absolute refractory period’ as new action potentials are relatively unlikely.

Propagation Down Unmyelinated Axons

Note that when a portion of the axon membrane is depolarized during an action potential, it may cause the excitation of neighbouring areas of the membrane. This might be thought to lead to chaotic, continual firings as portion A excites portion B which in turn excites portion A. However, the absolute refractory period means than regions that have recently fired are delayed in firing again. This results in a coherent wave propagating down the axon (analogous to a Mexican wave?) at a speed of 0.5…2.0 m/s = 1…4mph. This is slow – just walking pace!

Propagation Down Myelinated Axons: Saltatory Conduction

The myelin sheaths the axon with regular intervals which are unmyelinated, as

Different oligodendrocytes or Schwann cells myelinate different portions of an axon, with small gaps between them (nodes of Ranvier). Propagation down a myelinated portion will be much faster than an unmyelinated portion because of the increased parallel resistance and decreased parallel capacitance. The signal will be attenuated but, if it is still large enough when it reaches an unmyelinated node, this will trigger a new action potential to form at the node which will permit propagation of the signal quickly down the next myelinated stretch of the axon. And so the signal will continue, for as long as there is myelination.

This behavious is called salutatory conduction (from the Latin saltare, to hop or leap), so-called because the signal seems to hop from one node to the next.  It is much faster than  conduction along unmylinated axons. In the most thoroughly myelinated axons (12–20 μm in diameter), signal propagation is 70…120 m/s – around 200mph!).

Internodal lengths must be short enough for there to be sufficient excitation at the next node to trigger a new action potential. But too short an internodal length will result in slow signal propagation. Because attenuation depends on parallel resistance and capacitance, there is some relationship between  myelin sheath diameters and internodal lengths:

  • 1um external diameter => 50um length
  • 12um external diameter => 750um length

[model of myelinated cable]

Putting it all Together: Simulating Propagation down Axons

The above hopefully provides enough of a qualitative understanding of axon signal propagation to be able to use a quantitative simulation  tool.

Thomas Pollinger’s Java applet of the spread of excitation along an axon using the Hodgkin-Huxley model is something to play with. Note: try a different browser if it doesn’t work for you – it needs the Java virtual machine for Java 1.1 built in.

Mathematical modelling tools (such as Matlab) can be used for small-scale simulation and, with all the analogies with electronics, it should be apparent how it is possible to use a general analog electrical simulation tool such as SPICE to for larger-scale neuronal modelling, although more specialist software packages are now normally used.

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